Leon Brown

Cyclic vectors in the Dirichlet space

We study the Hilbert space of analytic functions with finite Dinchlet integral in the open unit disc. We try to identify the functions whose polynomial multiples are dense in this space. Theorems 1 and 2 confirm a special case of the following conjecture: if IJ(z)I > Ig(z)l at all points and if g is cyclic, thenJis cyclic. Theorems 3-5 give a sufficient condition (t is an outer function with some smoothness and the boundary zero set is at most countable) and a necessary condition (the radial limit can vanish only for a set of loganthmic capacity zero) for a function J to be cyclic. Introduction. In this paper we shall study the (Hilbert) space of analytic functions in the open unit disc a in the complex plane that have a finite Dirichlet integral: JJ If t12 dx dy [g(z)l for some cyclic g, and all z? Question 4 asks if f must be cyclic whenever f and l/f are both in the space. No examples are known where either of these questions has a negative answer. In §2 we begin the study of cyclic vectors in the Dirichlet space D. Theorems 1 and 2 give a partial answer to Question 3 above, for this space. This section also contains 2 propositions (10, 11) and 4 questions (7-10). Proposition 11 says that if f and g are bounded functions in D whose product is cyclic, then bothf and g must be cyclic. Theorem 2 gives a partial converse (we require that [gl be Dini continuous on Received by the editors July 21, 1983. 1980 Mathematics Subject Classification. Primary 30H05; Secondary 46E15, 46E20, 47B37.

Approximation and extension of random functions

Stochastic versions of the extension theorems of Tietze and Dugundji are obtained, as well as an existence theorem for partitions of unity by random continuous functions. A form of the classical approximation theorem of Mergelyan valid for random holomorphic functions on random compact sets is presented. A similar approach yields versions of the approximation theorems of Runge, Arakelyan, and Vitushkin.

Some examples of cyclic vectors in the Dirichlet space

We consider the Hilbert space of analytic functions in the open unit disc that have a finite Dirichlet integral. For E, a closed subset of the unit circle with logarithmic capacity zero, we construct a function in this space which is uniformly continuous, vanishes on E, and is cyclic with respect to the shift operator. For A, the open unit disk in the complex plane, let D be the space of functions f analytic on A with finite Dirichlet integral J f12 dx dy < oo. If f has a Taylor expansion

Measurable extension theorems

Abstract This paper concerns the use of measurable selection techniques to obtain some measurable extension theorems.

Some non-quasireflexive spaces having unique isomorphic preduals

It is shown that the dual spaces of certain James-Lindenstrauss spaces are spaces which are non-quasireflexive but have unique isomorphic preduals.

OUTER MEASURES ON A LINEAR LATTICE

In this paper we define and discuss the theory of abstract outer measures on a sequentially continuous(2) linear lattice S. This is a generalization of the concept of outer measure on a function space as used by Bourbaki [3]. H. Nakano [7] and M. H. Stone [8] have modernized Lebesgues extension theory; our approach provides a common generalization of their theories. We show, for example, that their theories lead to the same set of integrable functions, and that, from a lattice theoretic point of view, Stones results depend on the fact that the space of all real-valued functions on a set X is a perfect(3) lattice.

Uniqueness of preduals of certain Banach spaces

In [1], the authors have shown the existence of non-quasireflexive Banach spaces having unique isomorphic preduals. In fact, certain James-Lindenstrauss’ spaces have this property. In this paper it is shown that there are many such separable spaces. More precisely, there exist infinitely many different isomorphic types of James-Lindenstrauss’ spaces which are non-quasireflexive and have unique isomorphic preduals.